2-Probability Distributions & CIs.ipynb

{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"name":"Probability Distributions & CIs.ipynb","provenance":[],"collapsed_sections":[],"authorship_tag":"ABX9TyNPLDPH0j/ih+4KHOf0Y/JS"},"kernelspec":{"name":"python3","display_name":"Python 3"},"language_info":{"name":"python"}},"cells":[{"cell_type":"code","execution_count":null,"metadata":{"id":"iTsH7ZjkMj1a"},"outputs":[],"source":["import numpy as np\n","import pandas as pd\n","from scipy import stats\n","import seaborn as sns\n","import matplotlib.pyplot as plt"]},{"cell_type":"markdown","source":["**Binomial Distribution**\n","\n","**Exercise**\n","\n","The probability that a patient recovers from a stomach disease is 0.8.\n","\n","Suppose 20 people are known to have contracted this disease.\n","\n","What is the probability that\n","\n","a. exactly 14 recover?\n","\n","this is P(X=14)"],"metadata":{"id":"8BckzHtKMwBJ"}},{"cell_type":"code","source":["n = 20  #the number of trials\n","p = 0.8 #the probability of success\n","x = 14  #Random variable -Point probability is used"],"metadata":{"id":"KZBbCNeIMmEQ"},"execution_count":null,"outputs":[]},{"cell_type":"code","source":["stats.binom.pmf(k=14,n=20,p=0.8) #Probability mass function at k of the given Rv"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"EVtlalnAMmO6","executionInfo":{"status":"ok","timestamp":1657643074313,"user_tz":-180,"elapsed":318,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"83eb27bf-6495-47e1-ab84-bf0e5c04d67c"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.10909970097304983"]},"metadata":{},"execution_count":3}]},{"cell_type":"code","source":["stats.binom.mean(n,p) #the most expected value."],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"Gxg2-H90MmSF","executionInfo":{"status":"ok","timestamp":1657643164494,"user_tz":-180,"elapsed":289,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"f12cc6b8-a1da-4bc2-cc19-62d999b4871e"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["16.0"]},"metadata":{},"execution_count":4}]},{"cell_type":"markdown","source":["b. at least 10 recover?\n","\n","this is P(X≥10)=P(10)+P(11)+P(12)+...+P(20)"],"metadata":{"id":"9rkpdqpAPEGy"}},{"cell_type":"code","source":["x = 9           # selected for cdf value up to 9 is removed from all area."],"metadata":{"id":"6zoyBDAKMmT-"},"execution_count":null,"outputs":[]},{"cell_type":"code","source":["1-stats.binom.cdf(k=9,n=20,p=0.8)   # probability of at least 10 recover."],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"Yp72G0fIPrlT","executionInfo":{"status":"ok","timestamp":1657643415453,"user_tz":-180,"elapsed":340,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"1b080d0c-a739-43ec-fb00-7cfb93ac462b"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.9994365863023398"]},"metadata":{},"execution_count":6}]},{"cell_type":"markdown","source":["c. at least 14 but not more than 18 recover?\n","\n","this is P(14≤X≤18)=P(14)+P(15)+P(16)+P(17)+P(18)"],"metadata":{"id":"VykWrOYvQcNl"}},{"cell_type":"code","source":["x1 = 14\n","x2 = 18"],"metadata":{"id":"cyYpsmYoPrqx"},"execution_count":null,"outputs":[]},{"cell_type":"code","source":["#cdf method\n","stats.binom.cdf(18,20,0.8) - stats.binom.cdf(13,20,0.8) "],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"9re2xQ_mQjt-","executionInfo":{"status":"ok","timestamp":1657643717199,"user_tz":-180,"elapsed":288,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"080ea925-50db-4dca-9e0f-2671d486aa73"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.844132196156189"]},"metadata":{},"execution_count":9}]},{"cell_type":"code","source":["#pmf method\n","stats.binom.pmf(14,20,0.8)+stats.binom.pmf(15,20,0.8)+stats.binom.pmf(16,20,0.8)+stats.binom.pmf(17,20,0.8)+stats.binom.pmf(18,20,0.8)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"8jJ0DUtdQkQr","executionInfo":{"status":"ok","timestamp":1657644271310,"user_tz":-180,"elapsed":283,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"0cfe7595-bede-42fb-855e-b4f324a34ba0"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.8441321961561871"]},"metadata":{},"execution_count":11}]},{"cell_type":"markdown","source":["d. at most 16 recover?\n","\n","this is P(X≤16)=1−P(X>16)=P(17)+P(18)+P(19)+P(20)"],"metadata":{"id":"Ma6zOmWBTHqy"}},{"cell_type":"code","source":["x=16"],"metadata":{"id":"rEVUzPKSQkSv"},"execution_count":null,"outputs":[]},{"cell_type":"code","source":["stats.binom.cdf(16,20,0.8)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"VTEautf0TOiY","executionInfo":{"status":"ok","timestamp":1657644331522,"user_tz":-180,"elapsed":271,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"7fe7c354-3114-4ecf-8cac-bff5fe75d3e9"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.5885511380434305"]},"metadata":{},"execution_count":13}]},{"cell_type":"markdown","source":["e. Construct discrete probability distribution table and draw a graph"],"metadata":{"id":"KQZop6WfTru-"}},{"cell_type":"code","source":["dist = []\n","\n","for i in range(n+1):\n","    dist.append(stats.binom.pmf(i,n,p))\n","    print(str(i) + \"\\t\" + str(stats.binom.pmf(i,n,p)))"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"gGvliwzvTOlu","executionInfo":{"status":"ok","timestamp":1657644534819,"user_tz":-180,"elapsed":363,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"63020e27-ed04-45d2-9bd6-bfa1429ef396"},"execution_count":null,"outputs":[{"output_type":"stream","name":"stdout","text":["0\t1.048576000000001e-14\n","1\t8.388607999999975e-13\n","2\t3.187671040000001e-11\n","3\t7.650410495999974e-10\n","4\t1.3005697843199945e-08\n","5\t1.6647293239295892e-07\n","6\t1.664729323929588e-06\n","7\t1.3317834591436803e-05\n","8\t8.656592484433902e-05\n","9\t0.000461684932503141\n","10\t0.002031413703013826\n","11\t0.007386958920050259\n","12\t0.022160876760150803\n","13\t0.0545498504865252\n","14\t0.10909970097304983\n","15\t0.17455952155687976\n","16\t0.21819940194610007\n","17\t0.2053641430080944\n","18\t0.13690942867206304\n","19\t0.05764607523034236\n","20\t0.011529215046068481\n"]}]},{"cell_type":"code","source":["list(range(n+1))"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"tpHKNeiNTOnx","executionInfo":{"status":"ok","timestamp":1657644582154,"user_tz":-180,"elapsed":313,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"058285d7-3a2e-4e3f-cc81-5ed79a37c32c"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]"]},"metadata":{},"execution_count":15}]},{"cell_type":"code","source":["#matplotlib inline\n","plt.bar(list(range(n+1)), dist)\n","plt.show;"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":265},"id":"gbkspk7AURS-","executionInfo":{"status":"ok","timestamp":1657644619329,"user_tz":-180,"elapsed":387,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"1de66327-722c-4e67-b9f3-ed8f21cf58d4"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<Figure size 432x288 with 1 Axes>"],"image/png":"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\n"},"metadata":{"needs_background":"light"}}]},{"cell_type":"markdown","source":["**Poisson Distribution**\n","\n","**One parameter, success rare rate, distribution mean = lambda**\n","\n","Suppose that a random system of police patrol is devised so that a patrol officer may visit a given beat location Y = 0, 1, 2, 3, . . . times per half-hour period, with each location being visited an average of once per time period.\n","\n","Assume that Y possesses, approximately, a Poisson probability distribution.\n","\n","Calculate the probability that the patrol officer will miss a given location during a half-hour period.\n","\n","What is the probability that it will be visited once?\n","\n","Twice?\n","\n","At least once?"],"metadata":{"id":"cU10rHqrUl9n"}},{"cell_type":"code","source":["#For this example the time period is a half-hour, and the mean number of visits per half-hour interval is λ = 1. Then\n","\n","mu = 1  #rate"],"metadata":{"id":"t-WsX7SaURWv"},"execution_count":null,"outputs":[]},{"cell_type":"code","source":["stats.poisson.pmf(k=0,mu=1)    # P(Y = 0) = ?"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"SHafWBLgVx9W","executionInfo":{"status":"ok","timestamp":1657645142252,"user_tz":-180,"elapsed":401,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"0ded5f3a-3314-4c16-a00f-84e1200b256e"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.36787944117144233"]},"metadata":{},"execution_count":18}]},{"cell_type":"code","source":["stats.poisson.pmf(k=1,mu=1)    # P(Y = 1) = ?"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"mcz8brRjVyAr","executionInfo":{"status":"ok","timestamp":1657645280842,"user_tz":-180,"elapsed":298,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"933b2237-99d4-4d0d-eb07-24ee7aadd1f0"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.36787944117144233"]},"metadata":{},"execution_count":20}]},{"cell_type":"code","source":["stats.poisson.pmf(k=2,mu=1)    # P(Y = 2) = ?"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"Gf5fjRcrVyDQ","executionInfo":{"status":"ok","timestamp":1657645278948,"user_tz":-180,"elapsed":294,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"aeb1eac0-767b-4b64-aafd-b20e40859403"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.18393972058572114"]},"metadata":{},"execution_count":19}]},{"cell_type":"code","source":["  # P(Y ≥ 1) = ? #for discerete distrubiton only cdf and pmf are same.cdf 0 is removed from all.\n","\n","1-stats.poisson.cdf(k=0,mu=1)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"HrsnZDP0VyFi","executionInfo":{"status":"ok","timestamp":1657645407147,"user_tz":-180,"elapsed":344,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"5570dd7d-b905-4fb5-ef94-43490c97edb7"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.6321205588285576"]},"metadata":{},"execution_count":21}]},{"cell_type":"code","source":["1-stats.poisson.pmf(k=0,mu=1)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"rQ6u6ezlVyIB","executionInfo":{"status":"ok","timestamp":1657645452014,"user_tz":-180,"elapsed":293,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"4c6966ab-e484-45a1-a9f3-7ebf05c95790"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.6321205588285577"]},"metadata":{},"execution_count":22}]},{"cell_type":"code","source":["  #P(Y ≥ 3) = ?\n"," \n","1-stats.poisson.cdf(k=2,mu=1)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"lDwyl1oiYBUD","executionInfo":{"status":"ok","timestamp":1657645629286,"user_tz":-180,"elapsed":283,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"17b8a008-8a15-4e48-bec2-10ecb18c97f3"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.08030139707139416"]},"metadata":{},"execution_count":23}]},{"cell_type":"markdown","source":["**Normal Distribution**\n","\n","**A normal continuous random variable.Point-discerete probability is zero.\n","P(X=72)=0**\n","\n","Scores on an examination are assumed to be normally distributed with mean 78 and variance 36.\n","\n","a. What is the probability that a person taking the examination scores higher than 72?\n","\n","this is P(X>72)"],"metadata":{"id":"ak1UOZ_jYWJ4"}},{"cell_type":"code","source":["mu = 78          #mean\n","sd = np.sqrt(36) #standard deviation\n","x = 72           #random variable"],"metadata":{"id":"v_PkmnbPYBWz"},"execution_count":null,"outputs":[]},{"cell_type":"code","source":["1 - stats.norm.cdf(x=72, loc=78, scale=6)  # A normal continuous random variable.Measure a range possibility."],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"UFSnMuO5YBZv","executionInfo":{"status":"ok","timestamp":1657645967724,"user_tz":-180,"elapsed":308,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"4c5683bc-aeba-40a8-a36c-158dac93f2d7"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.8413447460685429"]},"metadata":{},"execution_count":25}]},{"cell_type":"markdown","source":["b. Suppose that students scoring in the top 10% of this distribution are to receive an A grade. What is the minimum score a student must achieve to earn an A grade?"],"metadata":{"id":"yiiTWtPmZjyE"}},{"cell_type":"code","source":["q = 1-0.1  # %90"],"metadata":{"id":"aUbcwt1gZGHw","executionInfo":{"status":"ok","timestamp":1657648973192,"user_tz":-180,"elapsed":282,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}}},"execution_count":26,"outputs":[]},{"cell_type":"code","source":["stats.norm.ppf(q) #corresponding z-score , default loc=0 scale=1"],"metadata":{"id":"H9liPRzvZGKt","colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"status":"ok","timestamp":1657649140739,"user_tz":-180,"elapsed":289,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"36c1cc3a-fcc3-4aaf-c696-5c87241c595c"},"execution_count":29,"outputs":[{"output_type":"execute_result","data":{"text/plain":["1.2815515655446004"]},"metadata":{},"execution_count":29}]},{"cell_type":"code","source":["stats.norm.ppf(q=0.90,loc=78,scale=6)  "],"metadata":{"id":"3H8qSh0WZGM_","colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"status":"ok","timestamp":1657649095660,"user_tz":-180,"elapsed":262,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"5400d4ef-b8e6-446f-af64-0f3280d86b18"},"execution_count":28,"outputs":[{"output_type":"execute_result","data":{"text/plain":["85.6893093932676"]},"metadata":{},"execution_count":28}]},{"cell_type":"markdown","source":["c. What must be the cutoff point for passing the examination if the examiner wants only the top 28.1% of all scores to be passing?"],"metadata":{"id":"ZAYXuH-Hl-2S"}},{"cell_type":"code","source":["q = 1-0.281"],"metadata":{"id":"VB8IeZjnlxb4","executionInfo":{"status":"ok","timestamp":1657649280678,"user_tz":-180,"elapsed":330,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}}},"execution_count":30,"outputs":[]},{"cell_type":"code","source":["stats.norm.ppf(q=1-0.281,loc=78,scale=6)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"BtGRCRsslxm2","executionInfo":{"status":"ok","timestamp":1657649323135,"user_tz":-180,"elapsed":363,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"ff5d273d-906a-4f5f-a309-e21efb2f551d"},"execution_count":31,"outputs":[{"output_type":"execute_result","data":{"text/plain":["81.47924035456623"]},"metadata":{},"execution_count":31}]},{"cell_type":"markdown","source":["d. Approximately what proportion of students have scores 5 or more points above the score that cuts off the lowest 25%?"],"metadata":{"id":"lEAKvLX1ma6G"}},{"cell_type":"code","source":["x = stats.norm.ppf(0.25,78,6)  # ppf goes possibility to score\n","x"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"kbVeCU3vmWfS","executionInfo":{"status":"ok","timestamp":1657649420227,"user_tz":-180,"elapsed":286,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"6a755c0b-d489-48d4-b822-ad988e8993dc"},"execution_count":33,"outputs":[{"output_type":"execute_result","data":{"text/plain":["73.95306149882352"]},"metadata":{},"execution_count":33}]},{"cell_type":"code","source":["1-stats.norm.cdf(x+5,78,6)     #cdf goes score to possibility"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"G6tGJoo_mWh7","executionInfo":{"status":"ok","timestamp":1657649447387,"user_tz":-180,"elapsed":290,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"632bdd98-1773-422f-fe55-ffcfbba6061a"},"execution_count":34,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.43689605516226615"]},"metadata":{},"execution_count":34}]},{"cell_type":"markdown","source":["f. If it is known that a student’s score exceeds 72, what is the probability that his or her score exceeds 84?"],"metadata":{"id":"KvBvkP7LndJQ"}},{"cell_type":"code","source":["# P(Y>84|Y>72) = P[Y>84 &(intercept) Y>72] / P(Y>72) = P(Y>84) / P(Y>72)  #conditional probability\n","\n","(1 - stats.norm.cdf(x=84, loc=78, scale=6)) / (1 - stats.norm.cdf(x=72, loc=78, scale=6))"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"75gB_v9XmWo4","executionInfo":{"status":"ok","timestamp":1657649803608,"user_tz":-180,"elapsed":289,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"f8e7623f-f2eb-4867-a62b-da04f82e4458"},"execution_count":35,"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.18857341734506025"]},"metadata":{},"execution_count":35}]},{"cell_type":"markdown","source":["**t Distribution**\n","\n","For a t distribution with 15 degrees of freedom, 90% of the distribution is within how many standard deviations of the mean?"],"metadata":{"id":"sotalOCoqCip"}},{"cell_type":"code","source":["df = 15   # t dist. parameter is degrees of freedom =n-1\n","q = 0.05  # q : lower tail probability"],"metadata":{"id":"hNJuADpfqRdL","executionInfo":{"status":"ok","timestamp":1657651695107,"user_tz":-180,"elapsed":389,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}}},"execution_count":41,"outputs":[]},{"cell_type":"code","source":["stats.t.ppf(q,df)  #from possibilty to score"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"zqcZv9JLoN8s","executionInfo":{"status":"ok","timestamp":1657651766912,"user_tz":-180,"elapsed":276,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"dbf0ef27-7236-4328-cea8-644ab98ed32a"},"execution_count":45,"outputs":[{"output_type":"execute_result","data":{"text/plain":["-1.7530503556925552"]},"metadata":{},"execution_count":45}]},{"cell_type":"markdown","source":["**Confidence Intervals Using the Normal Distribution**\n","\n","\n","If we’re working with larger samples (n≥30), we can assume that the sampling distribution of the sample mean is normally distributed (thanks to the Central Limit Theorem) and can instead use the norm.interval() function from the scipy.stats library."],"metadata":{"id":"ONVhlX0B7GMY"}},{"cell_type":"code","source":["sns.get_dataset_names()  #list of dataset names on seaborn"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"KRkBwV8ymWrl","executionInfo":{"status":"ok","timestamp":1657654838623,"user_tz":-180,"elapsed":733,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"881823e3-bda9-4063-ed11-ba0417147754"},"execution_count":46,"outputs":[{"output_type":"execute_result","data":{"text/plain":["['anagrams',\n"," 'anscombe',\n"," 'attention',\n"," 'brain_networks',\n"," 'car_crashes',\n"," 'diamonds',\n"," 'dots',\n"," 'exercise',\n"," 'flights',\n"," 'fmri',\n"," 'gammas',\n"," 'geyser',\n"," 'iris',\n"," 'mpg',\n"," 'penguins',\n"," 'planets',\n"," 'taxis',\n"," 'tips',\n"," 'titanic']"]},"metadata":{},"execution_count":46}]},{"cell_type":"code","source":["#Load \"tips\" dataset from Seaborn library\n","\n","tips = sns.load_dataset(\"tips\")\n","tips.head()"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":206},"id":"aOn-CYLNmWt8","executionInfo":{"status":"ok","timestamp":1657654981398,"user_tz":-180,"elapsed":873,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"d7802a96-8147-4928-ea8c-6549b055ec67"},"execution_count":47,"outputs":[{"output_type":"execute_result","data":{"text/plain":["   total_bill   tip     sex smoker  day    time  size\n","0       16.99  1.01  Female     No  Sun  Dinner     2\n","1       10.34  1.66    Male     No  Sun  Dinner     3\n","2       21.01  3.50    Male     No  Sun  Dinner     3\n","3       23.68  3.31    Male     No  Sun  Dinner     2\n","4       24.59  3.61  Female     No  Sun  Dinner     4"],"text/html":["\n","  <div id=\"df-c7d0ffde-dba4-4eb0-9d33-7c771abaa9ca\">\n","    <div class=\"colab-df-container\">\n","      <div>\n","<style scoped>\n","    .dataframe tbody tr th:only-of-type {\n","        vertical-align: middle;\n","    }\n","\n","    .dataframe tbody tr th {\n","        vertical-align: top;\n","    }\n","\n","    .dataframe thead th {\n","        text-align: right;\n","    }\n","</style>\n","<table border=\"1\" class=\"dataframe\">\n","  <thead>\n","    <tr style=\"text-align: right;\">\n","      <th></th>\n","      <th>total_bill</th>\n","      <th>tip</th>\n","      <th>sex</th>\n","      <th>smoker</th>\n","      <th>day</th>\n","      <th>time</th>\n","      <th>size</th>\n","    </tr>\n","  </thead>\n","  <tbody>\n","    <tr>\n","      <th>0</th>\n","      <td>16.99</td>\n","      <td>1.01</td>\n","      <td>Female</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>2</td>\n","    </tr>\n","    <tr>\n","      <th>1</th>\n","      <td>10.34</td>\n","      <td>1.66</td>\n","      <td>Male</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>3</td>\n","    </tr>\n","    <tr>\n","      <th>2</th>\n","      <td>21.01</td>\n","      <td>3.50</td>\n","      <td>Male</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>3</td>\n","    </tr>\n","    <tr>\n","      <th>3</th>\n","      <td>23.68</td>\n","      <td>3.31</td>\n","      <td>Male</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>2</td>\n","    </tr>\n","    <tr>\n","      <th>4</th>\n","      <td>24.59</td>\n","      <td>3.61</td>\n","      <td>Female</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>4</td>\n","    </tr>\n","  </tbody>\n","</table>\n","</div>\n","      <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-c7d0ffde-dba4-4eb0-9d33-7c771abaa9ca')\"\n","      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interval\n"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":279},"id":"-NUJG2NPmXWC","executionInfo":{"status":"ok","timestamp":1657655186209,"user_tz":-180,"elapsed":297,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"1772b019-2b26-4441-d919-0f685dfa7a59"},"execution_count":49,"outputs":[{"output_type":"display_data","data":{"text/plain":["<Figure size 432x288 with 1 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a dataframe just including Sunday tips\n","\n","tipsSun = tips[tips[\"day\"] == \"Sun\"]"],"metadata":{"id":"xnL2_GUb8aL6","executionInfo":{"status":"ok","timestamp":1657655615677,"user_tz":-180,"elapsed":269,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}}},"execution_count":50,"outputs":[]},{"cell_type":"code","source":["tipsSun['total_bill'].mean()"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"bKlumo24mXY_","executionInfo":{"status":"ok","timestamp":1657655728937,"user_tz":-180,"elapsed":275,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"379223b9-77d2-440d-cee5-5d85b88f77c8"},"execution_count":54,"outputs":[{"output_type":"execute_result","data":{"text/plain":["21.41"]},"metadata":{},"execution_count":54}]},{"cell_type":"code","source":["tipsSun[\"total_bill\"].std()"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"ywNZ-3gsmXbz","executionInfo":{"status":"ok","timestamp":1657655725478,"user_tz":-180,"elapsed":5,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"95ec4ea1-62a9-448d-fc35-d889b38b216a"},"execution_count":53,"outputs":[{"output_type":"execute_result","data":{"text/plain":["8.832121828869889"]},"metadata":{},"execution_count":53}]},{"cell_type":"code","source":["#Calculate standart error of the mean\n","\n","sem = tipsSun[\"total_bill\"].std() / np.sqrt(tipsSun.shape[0])\n","sem"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"npu0b6Ab-yWn","executionInfo":{"status":"ok","timestamp":1657656109367,"user_tz":-180,"elapsed":273,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"bfb9b91f-9157-4d80-9832-d35db66a1657"},"execution_count":61,"outputs":[{"output_type":"execute_result","data":{"text/plain":["1.0131138555021968"]},"metadata":{},"execution_count":61}]},{"cell_type":"code","source":["tipsSun"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":424},"id":"Y5PEhNc9-yb7","executionInfo":{"status":"ok","timestamp":1657656035753,"user_tz":-180,"elapsed":19,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"eb93f45b-378a-4925-856e-25b0a6c1ea74"},"execution_count":58,"outputs":[{"output_type":"execute_result","data":{"text/plain":["     total_bill   tip     sex smoker  day    time  size\n","0         16.99  1.01  Female     No  Sun  Dinner     2\n","1         10.34  1.66    Male     No  Sun  Dinner     3\n","2         21.01  3.50    Male     No  Sun  Dinner     3\n","3         23.68  3.31    Male     No  Sun  Dinner     2\n","4         24.59  3.61  Female     No  Sun  Dinner     4\n","..          ...   ...     ...    ...  ...     ...   ...\n","186       20.90  3.50  Female    Yes  Sun  Dinner     3\n","187       30.46  2.00    Male    Yes  Sun  Dinner     5\n","188       18.15  3.50  Female    Yes  Sun  Dinner     3\n","189       23.10  4.00    Male    Yes  Sun  Dinner     3\n","190       15.69  1.50    Male    Yes  Sun  Dinner     2\n","\n","[76 rows x 7 columns]"],"text/html":["\n","  <div id=\"df-8a37a1fc-394b-4039-bb9b-e5cf151d78d8\">\n","    <div class=\"colab-df-container\">\n","      <div>\n","<style scoped>\n","    .dataframe tbody tr th:only-of-type {\n","        vertical-align: middle;\n","    }\n","\n","    .dataframe tbody tr th {\n","        vertical-align: top;\n","    }\n","\n","    .dataframe thead th {\n","        text-align: right;\n","    }\n","</style>\n","<table border=\"1\" class=\"dataframe\">\n","  <thead>\n","    <tr style=\"text-align: right;\">\n","      <th></th>\n","      <th>total_bill</th>\n","      <th>tip</th>\n","      <th>sex</th>\n","      <th>smoker</th>\n","      <th>day</th>\n","      <th>time</th>\n","      <th>size</th>\n","    </tr>\n","  </thead>\n","  <tbody>\n","    <tr>\n","      <th>0</th>\n","      <td>16.99</td>\n","      <td>1.01</td>\n","      <td>Female</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>2</td>\n","    </tr>\n","    <tr>\n","      <th>1</th>\n","      <td>10.34</td>\n","      <td>1.66</td>\n","      <td>Male</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>3</td>\n","    </tr>\n","    <tr>\n","      <th>2</th>\n","      <td>21.01</td>\n","      <td>3.50</td>\n","      <td>Male</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>3</td>\n","    </tr>\n","    <tr>\n","      <th>3</th>\n","      <td>23.68</td>\n","      <td>3.31</td>\n","      <td>Male</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>2</td>\n","    </tr>\n","    <tr>\n","      <th>4</th>\n","      <td>24.59</td>\n","      <td>3.61</td>\n","      <td>Female</td>\n","      <td>No</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>4</td>\n","    </tr>\n","    <tr>\n","      <th>...</th>\n","      <td>...</td>\n","      <td>...</td>\n","      <td>...</td>\n","      <td>...</td>\n","      <td>...</td>\n","      <td>...</td>\n","      <td>...</td>\n","    </tr>\n","    <tr>\n","      <th>186</th>\n","      <td>20.90</td>\n","      <td>3.50</td>\n","      <td>Female</td>\n","      <td>Yes</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>3</td>\n","    </tr>\n","    <tr>\n","      <th>187</th>\n","      <td>30.46</td>\n","      <td>2.00</td>\n","      <td>Male</td>\n","      <td>Yes</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>5</td>\n","    </tr>\n","    <tr>\n","      <th>188</th>\n","      <td>18.15</td>\n","      <td>3.50</td>\n","      <td>Female</td>\n","      <td>Yes</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>3</td>\n","    </tr>\n","    <tr>\n","      <th>189</th>\n","      <td>23.10</td>\n","      <td>4.00</td>\n","      <td>Male</td>\n","      <td>Yes</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>3</td>\n","    </tr>\n","    <tr>\n","      <th>190</th>\n","      <td>15.69</td>\n","      <td>1.50</td>\n","      <td>Male</td>\n","      <td>Yes</td>\n","      <td>Sun</td>\n","      <td>Dinner</td>\n","      <td>2</td>\n","    </tr>\n","  </tbody>\n","</table>\n","<p>76 rows × 7 columns</p>\n","</div>\n","      <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-8a37a1fc-394b-4039-bb9b-e5cf151d78d8')\"\n","              title=\"Convert 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moe"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"4CMQ3L3TBPw1","executionInfo":{"status":"ok","timestamp":1657657065839,"user_tz":-180,"elapsed":4,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"3a682f7d-5b23-4f36-f7df-d43ceec07fae"},"execution_count":72,"outputs":[{"output_type":"execute_result","data":{"text/plain":["19.424296843215693"]},"metadata":{},"execution_count":72}]},{"cell_type":"code","source":["#Calculate 95% Confidence Interval around the mean by using scipy.stats\n","\n","stats.norm.interval(0.95, loc = tipsSun[\"total_bill\"].mean(), scale = tipsSun[\"total_bill\"].sem())"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"xnlhIBv8BQCQ","executionInfo":{"status":"ok","timestamp":1657657227849,"user_tz":-180,"elapsed":292,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"9a7fce3b-1ad4-43f6-e62d-52c4c193212a"},"execution_count":73,"outputs":[{"output_type":"execute_result","data":{"text/plain":["(19.424333330977177, 23.395666669022823)"]},"metadata":{},"execution_count":73}]},{"cell_type":"markdown","source":["**Exercise Large Samples**\n","\n","The shopping times of n = 64 randomly selected customers at a local supermarket were recorded.\n","\n","The average and variance of the 64 shopping times were 33  and 256 , respectively.\n","\n","Estimate μ, the true average shopping time per customer, with a confidence coefficient of 1 − α = .90.\n","\n","**CI = xbar +- z*s/np.sqrt(n)**"],"metadata":{"id":"FieNwwBSFNbN"}},{"cell_type":"code","source":["n = 64\n","xbar = 33\n","var = 256\n","ci = 0.90\n","s = np.sqrt(var)\n","\n","sem = s/np.sqrt(64)"],"metadata":{"id":"INIR_8IIBQEd","executionInfo":{"status":"ok","timestamp":1657657623971,"user_tz":-180,"elapsed":291,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}}},"execution_count":74,"outputs":[]},{"cell_type":"code","source":["#Using Normal Distribution\n","stats.norm.interval(0.90, loc = xbar, scale = sem)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"lvsFHYsMGA1K","executionInfo":{"status":"ok","timestamp":1657657650468,"user_tz":-180,"elapsed":283,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"7eca6f8f-5654-4e15-afe0-b8aa78bc30cc"},"execution_count":75,"outputs":[{"output_type":"execute_result","data":{"text/plain":["(29.710292746097053, 36.28970725390295)"]},"metadata":{},"execution_count":75}]},{"cell_type":"code","source":["#Using t Distribution\n","stats.t.interval(0.90, df=n-1, loc = xbar, scale = sem)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"YsDmw3bkGA4F","executionInfo":{"status":"ok","timestamp":1657657663150,"user_tz":-180,"elapsed":285,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"1ecd82b9-afeb-4c6f-f379-91fad34bf471"},"execution_count":76,"outputs":[{"output_type":"execute_result","data":{"text/plain":["(29.661195556984076, 36.338804443015924)"]},"metadata":{},"execution_count":76}]},{"cell_type":"markdown","source":["**Exercise Small Samples**\n","\n","A manufacturer of gunpowder has developed a new powder, which was tested in eight shells. The resulting muzzle velocities, in feet per second, were as follows:\n","\n","3005 2925 2935 2965\n","\n","2995 3005 2937 2905\n","\n","Find a 95% confidence interval for the true average velocity μ for shells of this type. Assume that muzzle velocities are approximately normally distributed.\n","\n","**xbar(sampleMean) +- T ( t for confidence level) * standarderror**"],"metadata":{"id":"NcSxT1FVGSWP"}},{"cell_type":"code","source":["x = [3005, 2925, 2935, 2965, 2995, 3005, 2937, 2905]"],"metadata":{"id":"wj2NifqOGNPT","executionInfo":{"status":"ok","timestamp":1657657932102,"user_tz":-180,"elapsed":7,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}}},"execution_count":77,"outputs":[]},{"cell_type":"code","source":["# Sample size\n","n = len(x)\n","\n","# Sample mean\n","xbar = np.mean(x)\n","\n","# Sample standard deviation\n","s = np.std(x,ddof=1)\n","\n","# Confidence level\n","cl = 0.95"],"metadata":{"id":"3NZ01d1dGNWh","executionInfo":{"status":"ok","timestamp":1657658163859,"user_tz":-180,"elapsed":303,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}}},"execution_count":81,"outputs":[]},{"cell_type":"code","source":["stats.t.interval(cl, df=n-1, loc=xbar, scale=s/np.sqrt(n))"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"9KEbn9jvGA72","executionInfo":{"status":"ok","timestamp":1657658166614,"user_tz":-180,"elapsed":275,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"aa7d3b9d-35f0-49ad-e826-003d01af90a3"},"execution_count":82,"outputs":[{"output_type":"execute_result","data":{"text/plain":["(2926.3202426713874, 2991.6797573286126)"]},"metadata":{},"execution_count":82}]},{"cell_type":"code","source":["#confidence multiplier\n","t = stats.t.ppf(0.975, df=n-1)\n","t"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"0wDrOnkuGA_A","executionInfo":{"status":"ok","timestamp":1657658443768,"user_tz":-180,"elapsed":279,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"a1f35d68-579f-42cc-dc6d-f67106d4ce23"},"execution_count":83,"outputs":[{"output_type":"execute_result","data":{"text/plain":["2.3646242510102993"]},"metadata":{},"execution_count":83}]},{"cell_type":"code","source":["#standard error of the mean\n","sem = s/np.sqrt(n)\n","sem"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"IWU66XRVJJVV","executionInfo":{"status":"ok","timestamp":1657658457207,"user_tz":-180,"elapsed":3,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"99d19c99-2eca-445b-f05a-8d7aeedce137"},"execution_count":84,"outputs":[{"output_type":"execute_result","data":{"text/plain":["13.820274961085254"]},"metadata":{},"execution_count":84}]},{"cell_type":"code","source":["#margin of error\n","moe = t*sem\n","moe"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"NzIC1U8QJJYA","executionInfo":{"status":"ok","timestamp":1657658474015,"user_tz":-180,"elapsed":282,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"0682e8d6-04b9-462f-9e62-e31becb2f452"},"execution_count":85,"outputs":[{"output_type":"execute_result","data":{"text/plain":["32.67975732861261"]},"metadata":{},"execution_count":85}]},{"cell_type":"code","source":["#lower confidence limit\n","low_limit = xbar - moe\n","low_limit"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"iMY3-Wo4JQmJ","executionInfo":{"status":"ok","timestamp":1657658493402,"user_tz":-180,"elapsed":274,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"f7fe68f6-a975-47e1-86aa-880d58a18095"},"execution_count":86,"outputs":[{"output_type":"execute_result","data":{"text/plain":["2926.3202426713874"]},"metadata":{},"execution_count":86}]},{"cell_type":"code","source":["#higher confidence limit\n","high_limit = xbar + moe\n","high_limit"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"oWCZSWOaJQos","executionInfo":{"status":"ok","timestamp":1657658501746,"user_tz":-180,"elapsed":321,"user":{"displayName":"Humeyra Armutlu","userId":"12643211298338865125"}},"outputId":"dbe53c5b-fb66-49f3-9504-244545698e4e"},"execution_count":87,"outputs":[{"output_type":"execute_result","data":{"text/plain":["2991.6797573286126"]},"metadata":{},"execution_count":87}]},{"cell_type":"markdown","source":["**Source:**\n","\n"," Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning."],"metadata":{"id":"6o5M2oXSJfli"}},{"cell_type":"code","source":[""],"metadata":{"id":"EZVupNHRJJau"},"execution_count":null,"outputs":[]}]}